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Sharlotte L. B. Kramer • Emily Retzlaff • Piyush Thakre
Johan Hoefnagels • Marco Rossi • Attilio Lattanzi
François Hemez • Mostafa Mirshekari • Austin Downey
Editors
Additive and Advanced Manufacturing, Inverse Problem Methodologies and Machine Learning and Data Science, Volume 4
Proceedings of the 2023 Annual Conference & Exposition on Experimental and Applied Mechanics
Editors
Sharlotte L. B. Kramer
Sandia National Laboratories Albuquerque, NM, USA
Piyush Thakre Dow Inc. Lake Jackson, TX, USA
Marco Rossi
Università Politecnica delle Marche Ancona, Italy
François Hemez
Lawrence Livermore National Laboratory Livermore, CA, USA
Austin Downey University of South Carolina Columbia, SC, USA
Emily Retzlaff
United States Naval Academy Annapolis, MD, USA
Johan Hoefnagels
Eindhoven University of Techno Eindhoven, Noord-Brabant, The
Attilio Lattanzi
California Institute of Technolog Pasadena, CA, USA
Mostafa Mirshekari
Carnegie Mellon University Pittsburgh, PA, USA
logy Netherlands y
ISSN 2191-5644ISSN 2191-5652 (electronic)
Conference Proceedings of the Society for Experimental Mechanics Series ISBN 978-3-031-50473-0ISBN 978-3-031-50474-7 (eBook) https://doi.org/10.1007/978-3-031-50474-7
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Preface
Additive and Advanced Manufacturing, Inverse Problem Methodologies and Machine Learning and Data Science represents one of five volumes of technical papers presented at the 2023 SEM Annual Conference and Exposition on Experimental and Applied Mechanics organized by the Society for Experimental Mechanics held on June 5–8, 2023. The complete proceedings also include volumes on: Advancement of Optical Methods in Experimental Mechanics, Dynamic Behavior of Materials, Fracture and Fatigue, Mechanics of Biological Systems and Materials, Mechanics of Composite and Multifunctional Materials, Residual Stress, Thermomechanics and Infrared Imaging, and Time-Dependent Materials.
Each collection presents early findings from experimental and computational investigations on an important area within Experimental Mechanics.
Mechanics of Additive and Advanced Manufactured Materials is an emerging area due to the unprecedented design and manufacturing possibilities offered by new and evolving advanced manufacturing processes and the rich mechanics issues that emerge. Technical interest within the Society spans several other SEM Technical Divisions such as: Composites, Hybrids and Multifunctional Materials, Dynamic Behavior of Materials, Fracture and Fatigue, Residual Stress, Time-Dependent Materials, and the Research Committee.
The topic of mechanics of additive and advanced manufacturing included in this volume covers design, optimization, experiments, computations, and materials for advanced manufacturing processes (3D printing, micro-and nanomanufacturing, powder bed fusion, directed energy deposition, etc.) with particular focus on mechanics aspects (e.g., mechanical properties, residual stress, deformation, failure, rate-dependent mechanical behavior, etc.).
The conference organizers thank the authors, presenters, and session chairs for their participation, support, and contribution to this very exciting area of experimental mechanics.
Albuquerque, NM, USASharlotte L. B. Kramer Annapolis, MD, USAEmily Retzlaff Lake Jackson, TX, USAPiyush Thakre Eindhoven, Noord-Brabant, The NetherlandsJohan Hoefnagels Ancona, ItalyMarco Rossi Pasadena, CA, USAAttilio Lattanzi Livermore, CA, USAFrançois Hemez Pittsburgh, PA, USAMostafa Mirshekari Columbia, SC, USAAustin Downey
Quantifying Residual Stresses Generated by Laser-Powder Bed Fusion of Metallic Samples 1 Pouria Khanbolouki, Rodrigo Magana-Carranza, Eann Patterson, Chris Sutcliffe, and John Lambros
Loading-Unloading Compressive Response and Energy Dissipation of Liquid Crystal Elastomers and Their 3D Printed Lattice Structures at Various Strain Rates 7 Bo Song, Dylan Landry, Thomas Martinez, Christopher Chung, Kevin Long, Kai Yu, and Chris Yakacki
Residual Stress Induced in Thin Plates During Additive Manufacturing
Eann A. Patterson, John Lambros, Rodrigo Magana-Carranza, and Christopher J. Sutcliffe
Investigating the Effects of Acetone Vapor Treatment and Post Drying Conditions on Tensile and Fatigue Behavior of 3D Printed ABS Components 17 Heechang Bae, Nicholas Blair, Matthew Michaelis, and Awlad Hossain
Mechanics of Novel Double-Rounded-V Hierarchical Auxetic Structure: Finite Element Analysis and Experiments Using Three-Dimensional Digital Image
Rajesh Kumar and Iniyan Thiruselvam Repeatability of Residual Stress in Replicate Additively
Christopher R. D’Elia, Daniel R. Moser, Kyle L. Johnson, and Michael R. Hill
Acoustic Nondestructive Characterization of Metal Pantographs for Material and Defect Identi fi
Silviya M. Boyadzhieva, Lea S. Kollmannsperger, Florian Gutmann, Thomas Straub, and Sarah C. L. Fischer
Rapid Prototyping of a Micro-Scale Spectroscopic System by Two-Photon Direct Laser Writing
Anthony Salerni and Cosme Furlong
Bioinspired Interfaces for Improved Interlaminar Shear Strength in 3D Printed
Umut Altuntas, Demirkan Coker, and Denizhan Yavas
Thermo-mechanical Characterization of High-Strength Steel Through Inverse Methods
Marco Rossi, Luca Morichelli, and Steven Cooreman
A Multi-testing Approach for the Full Calibration of 3D Anisotropic Plasticity Models via Inverse Methods
Attilio Lattanzi, Mattia Utzeri, Marco Rossi, and Dario Amodio
Finite Element Based Material Property Identification Utilizing Full-Field Deformation Measurements
Sreehari Rajan Kattil, Subramani Sockalingam, Michael A. Sutton, and Tusit Weerasooriya
Data-Driven Material Models for Engineering Materials Subjected to Arbitrary Loading Paths: Influence of the Dimension of the Dataset
Burcu Tasdemir, Vito Tagarielli, and Antonio Pellegrino
Data-Driven Methodology to Extract Stress Fields in Materials Subjected to Dynamic
Vijendra Gupta and Addis Kidane
Quantifying Residual Stresses Generated by Laser-Powder Bed Fusion of Metallic Samples
Pouria Khanbolouki, Rodrigo Magana-Carranza, Eann Patterson, Chris Sutcliffe, and John Lambros
Abstract We use numerical modeling to predict residual stresses and deformations of thin metallic structures manufactured by laser-powder bed fusion. The effect of L-PBF process on residual deformations of thin quasi-2D structures is expected to be more substantial and complex than for thicker/bulk or axisymmetric components. Two types of geometries are considered: a thin horizontal plate for residual force measurements and thin vertical plates for residual deformations and support removal experiments. In both cases knowledge of the initially deformed shape and internal residual stresses will affect experimental interpretation. The numerical scheme used (ANSYS Additive Suite) involves weakly coupled thermomechanical simulations in a commercially available finite element package. It is shown that the simulations are in qualitative and general quantitative agreement with the experimental measurements within this numerical framework. Additionally, it is shown that the provided numerical framework can be used to predict the effect of support removal sequence on the final geometry of thin metallic structures.
Additively manufactured parts made with the laser-powder bed fusion (L-PBF) process are susceptible to build defects associated with residual stresses during processing. For bulk objects, these residual stresses manifest themselves in the microstructure as stress/strain inhomogeneities at the grain level, as well as in the macroscale as dimensional tolerances of the manufactured part. While the platform that the parts are built on constrains their distortion during the build process, the removal of supports between part and platform (cut-off) activates additional permanent deformations and redistribution of residual stresses in the parts. Thin metallic structures on the other hand, have been seen to generate additional complexities during the build and removal process, requiring frames and lateral support structures [1]. These structures are less explored while their use is commonplace in industries such as aerospace. Additive manufacturing can reduce the material waste of traditional methods used for manufacturing these components [2]. Knowledge of the residual stresses in the final part is important not only for corrections of possible geometric tolerances necessary in the finished part, but also for an assessment of the part’s mechanical performance. In the past decade, many numerical schemes have been developed to predict the residual deformations and stresses of an additively manufactured part [3–8]. Only a handful of these studies investigated the residual stresses and deformations after the support removal steps [9]; and to the best of our knowledge, no studies were focused on the residual deformations of thin metallic plates through numerical methods. Additionally, the part removal for the built component in the mentioned studies incorporated a single-step plate removal scheme. Most often, the parts are removed from the build platform in a series of steps by cutting techniques. One objective of this study is to predict the residual forces
P. Khanbolouki (✉) · J. Lambros
Department of Aerospace Engineering, University of Illinois Urbana-Champaign, Champaign, IL, USA e-mail: pouria@illinois.edu
R. Magana-Carranza · E. Patterson
Department of Mechanical, Materials & Aerospace Engineering, University of Liverpool, Liverpool, UK
S. L. B. Kramer et al. (eds.), Additive and Advanced Manufacturing, Inverse Problem Methodologies and Machine Learning and Data Science, Volume 4, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-50474-7_1
and deformations in additively manufactured quasi-2D structures. Another objective of this study is to investigate whether the sequence with which the supports are removed affects the redistribution of residual strains in the additively manufactured parts.
Background
In recent experimental studies by Magana-Carranza et al. horizontal plates were manufactured with L-PBF process for the in-situ force measurement of additive manufacturing process to infer the development of stresses [10, 11]. A four-by-four array of force transducer devices (FTD) were implemented and the support-structures were built on top of the connecting rods on the strain gauge load cells. In another experimental study, the additive manufacturing of vertical thin plates in portrait and landscape mode were explored and challenges of additive manufacturing of thin plates were discussed [1]. Frames and lateral triangular-shaped support buttresses were added to prevent the failure of the plates during the build process, indicating the additional complexities in additive manufacturing of quasi-2D structures. The measurements included the out-of-plane deformations of the plates at the end of the build process as well every support removal step. In both cases, knowledge of the initially deformed shape and internal residual stresses will affect experimental interpretation. Here, both types of geometries are considered: horizontal plates for in-situ force measurements during the build process and reinforced vertical thin plates for residual deformation redistributions during the cut-off process. The numerical scheme used (ANSYS Additive Suite) involves weakly coupled thermomechanical simulations in a commercially available FE package (Fig. 1). In this scheme, a transient thermal history analysis is performed on the undeformed mesh in a layer lumping technique, the results of which are utilized as input for a static mechanical simulation.
Analysis
As mentioned, two types of geometries are considered for this study: horizontal plates for in-situ force measurements during the build process and reinforced vertical thin plates for residual deformation redistributions during the cut-off process. For the horizontal plate simulation, a quarter symmetry design was implemented to the CAD model to reduce the computational costs. Similar to the experiments and previous research, the calculated forces developed in the central region of the part were compressive, while the corners undergo tension. The experimental measurements collected from FTDs during the build process are shown in Fig. 2 Additionally, comparable values to the experimental measurement for residual forces at the center
Fig. 1 Numerical scheme for simulation of L-PBF process of thin metallic structures in the commercial package ANSYS
Fig. 2 Forces measured by each FTD versus time/layer number during the build process of the square flat plate built with Inconel 625 and the arrangement of FTDs for in situ force measurements during build process of this square flat plate
Fig. 3 Finite element analysis results after the L-PBF part is removed from the build platform, presenting the calculated directional deformations of the part and the supports in the build direction (z-axis)
and corners of the thin horizontal plate were acquired from the simulations. Figure 3 presents the results of calculated directional deformations of the thin plate and the supports in the build direction (along z-axis).
The second set of simulations were included to compare the final deformed shaped of thin plates after cooling down and the removal steps (Fig. 4). The symmetrical design of the plates as well as the sequence in which the supports were removed, aimed at designing half of the plate for the simulations. The lateral supports are removed after the cool-down step, from both sides, followed by incremental removal of the bottom supports in 5 mm increments from either side toward the center of the plate. Additionally, full plates (without plane symmetry) were simulated to investigate the effect of support removal sequence
Fig. 4 Simulation progression for LPBF process including layer-by-layer deposition, cool-down, and support removal
of the vertical thin plates. Results from the simulations were post-processed in a third-party software (Origin pro and MATLAB) to acquire a one-to-one comparison at each step. The resulting calculated geometries from FEA were compared to stereo digital image correlation (stereo-DIC) measurements of manufactured specimens, before, during, and after the cut-off process for the vertical plates. The results from the simulations are in qualitative agreement with the 3D-DIC measurements. The effect of different cut-off sequences was explored on the residual deformations of the vertical plates. It was observed that the cut-off sequence might introduce additional deformations in quasi-2D structures.
Conclusion
It has been shown that the simulations are capable of providing acceptable qualitative and generally quantitative predictions of residual forces, residual deformations, and final geometries of thin structures. We can take advantage of these capabilities in developing optimization approaches for minimizing the residual deformations of thin metallic structures that are manufactured by L-PBF. The results also suggest that the cut-off sequence of the support structures of high aspect ratio components can result in substantial additional deformations in the part.
Acknowledgements The research was supported by grants from both the EPSRC (Grant No. EP/T013141/1) in UK and NSF CMMI (Grant No. 20–27082) in the USA. The opinions expressed in this article reflect only the authors’ view and EPSRC is not responsible for any use that may be made of the information it contains.
References
1. Patterson, E.A., Lambros, J., Magana-Carranza, R., Sutcliffe, C.J.: Residual stress effects during additive manufacturing of reinforced thin nickel–chromium plates. Int. J. Adv. Manuf. Technol. 123, 1845–1857 (2022)
2. Blakey-Milner, B., Gradl, P., Snedden, G., Brooks, M., Pitot, J., Lopez, E., Leary, M., Berto, F., du Plessis, A.: Metal additive manufacturing in aerospace: a review. Mater. Des. 209, 110008 (2021)
3. Li, C., Liu, Z.Y., Fang, X.Y., Guo, Y.B.: Residual stress in metal additive manufacturing. Procedia CIRP. 71, 348–353 (2018)
4. Kruth, J., Deckers, J., Yasa, E., Wauthlé, R.: Assessing and comparing influencing factors of residual stresses in selective laser melting using a novel analysis method. Proc. Inst. Mech. Eng. B J. Eng. Manuf. 226(6), 980–991 (2012)
5. Li, C., Guo, Y., Fang, X., Fang, F.: A scalable predictive model and validation for residual stress and distortion in selective laser melting. CIRP Ann. 67(1), 249–252 (2018)
6. Setien, I., Chiumenti, M., Sjoerd van der Veen, M.S.S., Garciandía, F., Echeverría, A.: Empirical methodology to determine inherent strains in additive manufacturing. Comput. Math. Appl. 78(7), 2282–2295 (2019)
7. Li, C., Liu, J.F., Fang, X.Y., Guo, Y.B.: Efficient predictive model of part distortion and residual stress in selective laser melting. Addit. Manuf. 17, 157–168 (2017)
8. Yakout, M., Elbestawi, M.A., Veldhuis, S.C., Nangle-Smith, S.: Influence of thermal properties on residual stresses in SLM of aerospace alloys. Rapid Prototyp. J. 26(1), 213–222 (2020)
9. Li, C., Liu, Z.Y., Fang, X.Y., Guo, Y.B.: On the simulation scalability of predicting residual stress and distortion in selective laser melting. J. Manuf. Sci. Eng. 140(4), 041013 (2018)
10. Carranza, R.M., Robinson, J., Ashton, I., Fox, P., Sutcliffe, C., Patterson, E.: A novel device for in-situ force measurements during laser powder bed fusion (L-PBF). Rapid Prototyp. J. 27(7), 1423–1431 (2021)
11. Magana-Carranza, R., Sutcliffe, C.J., Patterson, E.A.: The effect of processing parameters and material properties on residual forces induced in Laser Powder Bed Fusion (L-PBF). Addit. Manuf. 46, 102192 (2021)
Loading-Unloading Compressive Response andEnergyDissipation of Liquid Crystal Elastomers and Their 3D PrintedLattice Structures at Various Strain Rates
Bo Song, Dylan Landry, Thomas Martinez, Christopher Chung, Kevin Long, Kai Yu, and Chris Yakacki
Abstract Nematic liquid crystal elastomers (LCEs) are a unique class of network polymers with potential for excellent mechanical energy absorption and dissipation capacity due to their ability to change the nematic director under mechanical loading (sometimes called soft-elasticity) in addition to the viscoelastic behavior of the remaining polymer network. This additional inelastic mechanism makes them appealing as candidate damping materials in a variety of applications from vibration to impact. The lattice structures made from the LCEs provide further mechanical energy absorption and dissipation capacity associated with packing out the porosity.
Understanding the extent of mechanical energy absorption versus dissipation depends on the mechanical stress-strain response under both loading and unloading. In the past, the loading-unloading stress-strain response was only obtained within quasi-static (slow) strain rates on standard material test frames. In this study, we used a newly developed bench-top linear actuator to characterize the loading-unloading compressive response of polydomain and monodomain LCE polymers and polydomain LCE lattice structures with two different porosities (nominally, 62% and 85%) at both low and intermediate strain rates at room temperature. As a reference material, a bisphenol A (BPA) polymer with a similar glass transition temperature (9 °C) as the nematic LCE (4 °C) was also characterized at the same conditions for comparing to the LCE polymers. Based on the loading-unloading stress-strain curves, the energy absorption and dissipation for each material at different strain rates (0.001, 0.1, 1, 10 and 90 s-1) were able to be calculated. The strain-rate effect on the mechanical response and energy absorption and dissipation behaviors was determined.
Liquid crystal elastomers (LCEs) are a unique class of polymers that have multiple inelastic deformation mechanisms that confer unusual rate-dependent mechanical behavior, which makes them particularly attractive materials for damping and mechanical impact mitigation applications and potentially in biological or biomedical applications. In the LCEs studied here, liquid crystals (mesogen) are covalently bonded into a flexible polymer network to enable unique mechanical, thermal, and photo-mechanical properties, which make it be used as cell scaffolds, arti ficial muscles, interbody fusion cage, soft-active robotics or actuators, and wearable devices [1–6].
Recent studies of LCEs have investigated their unusual energy dissipation capacity due to their ability to undergo mesogen re-orientation under mechanical loading in addition to background viscoelastic effects [7–9]. Foam or lattice structures made from the LCEs have also been proposed for better energy dissipation and shock mitigation performance as compressive loading must first pack out the porosity before the mesogen re-orientation begins [10–12]. Understanding the energy dissipation characteristics depends on the mechanical stress-strain response under both loading and unloading. Considering the anticipated mechanical shock mitigation applications, strain rate becomes a critical variable for strain-rate-dependent hysteretic stress-strain response of the LCEs with both soft elastic and viscoelasticity mechanisms. However, the study of
B. Song (✉) · D. Landry · T. Martinez · K. Long
Sandia National Laboratories, Albuquerque, NM, USA
S. L. B. Kramer et al. (eds.), Additive and Advanced Manufacturing, Inverse Problem Methodologies and Machine Learning and Data Science, Volume 4, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-50474-7_2
strain-rate effects on the stress-strain response and energy dissipation of the LCEs and LCE-based lattice structures is very limited [8–14].
In this study, a high-speed bench-top linear actuator was employed to characterize the loading-unloading compressive response of polydomain and monodomain LCE polymers and polydomain LCE lattice structures with two different porosities (nominally, 62% and 85%) at low and intermediate strain rates. It is noted that the actual porosities for the two lattice structures were 50.8% and 73.7%, which are ~11% lower than the nominal values. As a reference material, a bisphenol A (BPA) polymer with a similar glass transition temperature (9 °C) as the nematic LCE (4 °C), both as characterized by the peak of the loss to storage modulus ratio in a 1 Hz three-point-bend dynamic mechanical analysis test swept at 3 °C/min, was also characterized at the same conditions for comparing to the LCE polymers. It is noted that, for the monodomain LCE polymer, the loading direction was along the direction of the nematic director. Based on the loading-unloading stress-strain curves, the energy absorption and dissipation for each material at different strain rates (0.001, 0.1, 1, 10 and 90 s-1) were calculated. The strain-rate effect on the mechanical behavior and energy dissipation was determined and compared between materials.
Experiments and Results
The compressive experiments on all five materials were conducted with a bench-top high-speed linear actuator that is presented in detail in the Journal of Dynamic Behavior of Materials [15]. As shown in Fig. 1, the test system consists of a Rexroth® high-speed electromechanical actuator to which a front platen was connected via an adapting rod. Another identical platen was installed on a Kistler load cell mounted on a back plate. The Kistler load cell was used to measure the force history on the specimen during mechanical loading. A customized laser extensometer [16] was installed to directly measure the displacement of actuator such that the specimen strain can be calculated. The compressive stress-strain curve of the material under investigation was then obtained. The linear actuator has a maximum velocity of ~1.9 m/s with a closed-loop operation mode, enabling loading-unloading stress-strain characterization at the strain rates up to 100 s-1 for the specimen design in this study. The signals from the load cell and the laser extensometer were acquired with a LeCroy digital oscilloscope.
The loading and unloading stress-strain curves of the five materials are shown in Fig. 2, respectively. It is noted that, due to significant strain-rate effect, the resultant stress-strain curves were plotted in two figures with the results at 1 s-1 used as a divide to ensure sufficient resolution within the entire strain-rate range. The results show that the polydomain LCE exhibited a very similar stress-strain response to the reference material BPA. A plateau was observed in the stress-strain curves of the monodomain LCE due to the effect of mesogen re-orientation and alignment (sometimes referred to soft-elasticity), which indicates that the monodomain LCE may have superior energy absorption to the polydomain LCE and BPA. The polydomain LCE lattice structures exhibited typical compressive stress-strain response of foam materials. With increasing porosity (85% vs. 62%), the stress at a certain strain significantly decreased.
Based on the compressive loading and unloading stress-strain curves shown in Fig. 2, the energy dissipation ratio was calculated with the following equation,
Fig. 1 Photograph of the high-speed linear actuator
2 Loading and unloading stress-strain curves of all five materials at various strain rates
Fig.
where loadingσ dε and unloadingσ dε are the integrals of the stress-strain curves for loading and unloading, respectively. Figure 3 compares the energy dissipation ratios at various strain rates for the five materials. The BPA and polydomain LCE had very similar strain rate effect on the energy dissipation ratio. At the strain rate of 0.001 s-1, both materials had energy dissipation ratios around 0.8. When the strain rate increased to 0.1 s-1, the energy dissipation ratios decreased to 0.32 and ~0.5 for the BPA and polydomain LCE, respectively. With further increase of strain rate, the energy dissipation ratios for the BPA and polydomain LCE increased. The monodomain LCE showed similar trend but the lowest energy dissipation ratio (~0.85) at the strain rate of 1 s-1. All BPA, polydomain LCE, and monodomain LCE had a very close energy dissipation ratio (~0.90) at the strain rate of 90 s-1, which suggests that viscoelasticity becomes dominant over the nematic director reorientation mechanism at higher strain rates (since all materials have similar Tgs and approximately similar viscoelastic behavior at room temperature). Although the porosities were different, the polydomain LCE lattice structures show very similar characteristic of energy dissipation ratio. When strain rate was below 1 s-1, the energy dissipation ratio was the same (~0.90) for both polydomain LCE lattice structures, indicating minimal strain rate effect. When strain rate is above 1 s-1, the energy dissipation ratio increased with increasing strain rate. At the strain rate of 90 s-1, nearly 100% energy was dissipated in the polydomain LCE lattice structures, which indicates that the polydomain LCE lattice structures had an excellent performance to mitigate external shock and impact. It is noted that, at the strain rate of 0.001 s-1, the monodomain LCE had the highest energy dissipation ratio among the materials including the lattice structures. This suggests that nematic director re-orientation dominates the material response at low strain rates, but still warrant further investigation.
Conclusion
3D printed polydomain LCE, monodomain LCE, and two polydomain LCE lattice structures with 62% and 85% porosities were mechanically characterized in compression with a high-speed linear actuator. A BPA polymer was also characterized as a reference material. Compressive loading and unloading hysteretic stress-strain curves for all five materials were obtained at various strain rates from 0.001 to 90 s-1. All five materials showed significant strain rate effect. Energy dissipation ratio was calculated from the resultant loading and unloading stress-strain curves. All five materials showed signi ficant but different strain rates on energy dissipation ratio. In general, the solid LCE and BPA materials showed great energy dissipation capabilities at both low (0.001 s-1) and high (above 1 s-1) strain rates, but not at the strain rates in between. The polydomain LCE lattice structure showed superior energy dissipation performance compared with the solid polymers.
Fig. 3 Energy dissipation ratio of the five materials at various strain rates
Acknowledgements Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525. The views expressed in the article do not necessarily represent the views of the U.S. Department of Energy or the United States Government.
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12. Mistry, D., Traugutt, N.A., Sanborn, B., Volpe, R.H., Chatham, L.S., Zhou, R., Song, B., Yu, K., Long, K.N., Yakacki, C.M.: Soft elasticity optimizes dissipation in 3D-printed liquid crystal elastomers. Nat. Commun. 12, 6677 (2021)
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Residual Stress Induced in Thin Plates DuringAdditive Manufacturing
Eann A. Patterson, John Lambros, Rodrigo Magana-Carranza, and Christopher J. Sutcliffe
Abstract Additive manufacturing is a technique for producing complex geometry engineering parts relatively quickly and cheaply; however, residual stresses induced in the part during manufacture can result in significant distortion of the build. In this study, nickel-chromium alloy (Inconel 625) geometrically-reinforced thin plates have been additively manufactured using laser-powder bed fusion, that have comparable flatness to those built subtractively. The residual stresses induced in the thin plates from manufacture are deduced by measuring out-of-plane displacements using stereoscopic digital image correlation. The results demonstrate that residual stresses cause potentially severe out-of-plane displacements which can be alleviated by using buttress supports to reinforce the plate edges during the build. In both landscape and portrait orientation builds, out-ofplane displacement increased upon release from the baseplate but was reduced by incremental release.
Thin plates with reinforced edges have potential applications as skins for divertors in fusion reactors and for hypersonic flight vehicles [1–3]. Subtractive manufacturing of such parts can create signi ficant waste and costly tool degradation for high temperature and corrosion resistant materials, such as Inconel.
Additive manufacturing (AM) is a modern technique for producing engineering components in a variety of materials in a layer-by-layer process based on the digital representation of the part. A key advantage of AM is the capability to produce complex geometry parts without the need for part-speci fic tools; however, when parts are created layer-by-layer during AM, differential strains caused by non-uniform plastic deformation and time-varying temperature distributions, induce macroscale residual stresses that can cause large distortions of the part [4]. The direct measurement of residual stress is difficult and instead residual stresses are usually deduced from measurements of displacements or strains induced by releasing residual stresses after build completion, employing techniques such as X-ray and neutron diffraction [5, 6]. Magana-Carranza et al. [7, 8] also evaluated residual stresses induced by AM by incorporating a force transducer into a laser-powder bed fusion (L-PBF) AM machine to measure forces induced during the build process.
In this study, nickel-chromium alloy (Inconel 625) geometrically-reinforced thin plates were additively manufactured using L-PBF and the residual stresses deduced from full-field stereoscopic digital image correlation (DIC) measurements of surface strains. Information has been obtained from plates built in both landscape and portrait orientations and with various build support structures. The study has resulted in a series of findings relating to build supports, orientation and support removal procedures, that minimise the distortion of the final plate.
E. A. Patterson (✉) · R. Magana-Carranza
Department of Mechanical, Materials & Aerospace Engineering, University of Liverpool, Liverpool, UK
e-mail: eannp@liverpool.ac.uk
J. Lambros
Department of Aerospace Engineering, University of Illinois Urbana-Champaign, Urbana-Champaign, IL, USA
S. L. B. Kramer et al. (eds.), Additive and Advanced Manufacturing, Inverse Problem Methodologies and Machine Learning and Data Science, Volume 4, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-50474-7_3
Methods
1 mm thick nickel-chromium alloy (Inconel 625) flat plates with in-plane dimensions of 130 mm × 230 mm surrounded by a reinforced frame of 10 mm by 5 mm, were built using a L-PBF machine (Renishaw AM250, UK). In all builds, the laser power was 400 W, the point distance 70 μm, and the layer thickness 60 μm; values that were selected based on prior experience [9]. A typical portrait orientation plate build, with support structures, is shown in Fig. 1a. The parts were built on a 170 °C pre-heated standard base plate for the machine. Preliminary tests confirmed that a stripe scan strategy produced lower levels of deformation, so this was implemented in all plate builds.
The shapes of the plates were measured using a stereoscopic DIC correlation system (Q400, Dantec Dynamics GmbH, Ulm, Germany). The system was setup as shown in Fig. 1b to achieve a spatial resolution of 20 pixels/mm using a pair of identical CCD cameras with 1292 × 964 pixels and 50 mm lenses. Each specimen was painted black at the end of its build process and a white speckle pattern applied to allow for DIC analysis. The equipment was calibrated before each set of measurements to determine the measurement uncertainty which typically had a mean of zero and a standard deviation of 0.0014 mm (see Fig. 1c).
After completion of the build process and specimen paint preparation, the base plate with the reinforced plate attached was placed on an optical table for DIC measurements. The supports were released incrementally alternating between each end of the geometry with stereoscopic images taken at each increment to allow the evolution of shape to be evaluated.
(c
DIC map
Fig. 1 (a) Exemplar of a successful L-PBF build of an Inconel 625 thin plate before base plate release (dimensions in mm); (b) Schematic diagram of the setup used for stereoscopic digital image correlation (DIC) measurements of out-of-plane displacement;
)
of measurement uncertainty. (Modified from Patterson et al. [4])
Results and Discussion
An out-of-plane step occurred in a number of the initial builds in the landscape orientation at a height of around 105 mm as illustrated in Fig. 2a, prompting trials with different support structures to prevent its occurrence. An enveloping structure successfully produced a plate with a flatness of 5.05 mm but created signi ficant material waste which negated any advantage of using AM. Metrologically flatness is defined as the minimum distance between two planes within which all the points on a surface lie [10]. Successful landscape builds, with minimal material wastage, were produced when triangular support buttresses orientated perpendicular to the plane of the reinforced plate were used. Although the builds in landscape orientation with triangular supports were successful, they produced significant out-of-plane displacement and a flatness of 5.7 mm when released from the support structures. An alternate build approach was sought, so it was decided to build the plates in a portrait orientation with triangular support buttresses orientated perpendicular to the plane of the reinforced plate. This strategy was unsuccessful and resulted in a horizontal discontinuity at a height of 113 mm and a delamination at 225 mm as shown in Fig. 2b. The addition of in-plane buttresses removed the delamination and discontinuity, resulting in a successful build (Fig. 1a) that when released from the base plate, resulted in a flatness of 4.6 mm as shown in Fig. 3a; an improved flatness of approximately 9% when compared with the build in landscape orientation with the envelope supports. The difference between displacement in the plates built in landscape and portrait orientations, likely results from the geometric effects associated with constraining the long and short edges to the base plate; as well as from differential thermal strains between layers in the transverse and longitudinal directions.
The formation of residual stresses and consequential deformation of the plates are time-varying processes and historydependent. They occur when the balance of forces in the plate changes during the build process due to the addition of mass to the part as well as due to energy transfers from the laser and to the surroundings. This balance of forces within the plate also changes during the removal of the plate from the baseplate and the supporting structures. The sequence of out-of-plane displacement fields shown in Fig. 3b for a reinforced plate built in the portrait orientation, illustrates this changing state of the forces during release from the baseplate. Each step shows the changes to the shape of the part as the supports are released in 5 mm increments. Figure 3b shows that the displacements are substantially larger in the bottom half of the plate when compared to the top (also observed for the builds with the landscape orientation). This result was anticipated as the effect of the residual forces reacted through the base plate would be expected to be greater adjacent to the baseplate and to dissipate with distance from the baseplate.
For builds that use other aspect ratios or materials, it seems likely that the use of buttress supports would be effective in reducing out-of-plane displacement. Although it is hard to predict this behaviour from the data collected in this study, the datasets should be invaluable for the development and validation of computational models that could provide such predictions of residual stress development in AM builds.
Fig. 2 (a) Example of a failed build in the landscape orientation; (b) Example of a failed build in the portrait orientation. (Modified from Patterson et al. [4])
Fig. 3 (a) Measurements of out-of-plane displacement, before and after baseplate release, for a build in the portrait orientation with both in-plane and out-of-plane buttresses; (b) Out-of-plane displacements measured at increments of 5 mm release from the baseplate for the first 12 steps and final 4 steps. (Modified from Patterson et al. [4])
Conclusion
Geometrically reinforced thin plates have been successfully built using laser-powder bed fusion (L-PBF) and their shapes measured using stereoscopic DIC before, during and after release from the baseplate of the AM machine. The results demonstrate that residual stresses induced by AM cause severe out-of-plane deformation that can be minimised by using in-plane and out-of-plane support buttresses. The out-of-plane displacement that occurs during release from the base plate can be minimised by incremental releases of 5 mm. Plates built in the portrait orientation with both in-plane and out-of-plane support buttresses resulted in builds with an out-of-plane deformation of 4.6 mm; the lowest out-of-plane displacement for all geometries built in this study.
Acknowledgements The research was supported by grants from both the EPSRC (Grant No. EP/T013141/1) in UK and NSF CMMI (Grant No. 20–27082) in the USA. The opinions expressed in this article reflect only the authors’ view and EPSRC is not responsible for any use that may be made of the information it contains. The authors are grateful to funders for providing the resources for the research and to the University of Liverpool for access to facilities to perform the research.
References
1. Santos Silva, A.C., Sebastian, C.M., Lambros, J., Patterson, E.A.: High temperature modal analysis of a non-uniformly heated rectangular plate: experiments and simulations. J. Sound Vib. 443, 397–410 (2019)
2. Lopez-Alba, E., Sebastian, C.M., Santos Silva, A.C., Patterson, E.A.: Experimental study of mode shifting in an asymmetrically heated rectangular plate. J. Sound Vib. 439, 241–250 (2019)
3. Silva, A.C.S., Lambros, J., Garner, D.M., Patterson, E.A.: Dynamic response of a thermally stressed plate with reinforced edges. Exp. Mech. 60(1), 81–92 (2020)
4. Patterson, E.A., Lambros, J., Magana-Carranza, R., Sutcliffe, C.J.: Residual stress effects during additive manufacturing of reinforced thin nickel–chromium plates. Int. J. Adv. Manuf. Technol. 123(5), 1845–1857 (2022)
5. Mercelis, P., Kruth, J.-P.: Residual Stresses in Selective Laser Sintering and Selective Laser Melting. Emerald Group Publishing Limited (2006)
6. Qian, W., Wu, S., Wu, Z., Ahmed, S., Zhang, W., Qian, G., Withers, P.J.: In situ X-ray imaging of fatigue crack growth from multiple defects in additively manufactured AlSi10Mg alloy. Int. J. Fatigue. 155, 106616 (2022)
7. Magana Carranza, R., Robinson, J., Ashton, I., Fox, P., Sutcliffe, C., Patterson, E.: A novel device for in-situ force measurements during laser powder bed fusion (L-PBF). Rapid Prototyp. J. 27(7), 1423–1431 (2021)
8. Magana-Carranza, R., Sutcliffe, C.J., Patterson, E.A.: The effect of processing parameters and material properties on residual forces induced in Laser Powder Bed Fusion (L-PBF). Addit. Manuf. 46, 102192 (2021)
9. Bartlett, J., Li, X.: An Overview of Residual Stresses in Metal Powder Bed Fusion, vol. 27, pp. 131–149 (2019)
10. Thomas, G.G.: Engineering metrology. Butterworths, London (1974)
Investigating the Effects of Acetone Vapor TreatmentandPost Drying Conditions on Tensile
and Fatigue Behaviorof3DPrinted ABS Components
Heechang Bae, Nicholas Blair, Matthew Michaelis, and Awlad Hossain
Abstract Fused Deposition Modeling (FDM), an additive manufacturing/3D printing process, is widely used where the material is melted, extruded, and deposited in layers to build up the desired object. The applications of FDM technologies have significantly increased recently not only for rapid prototyping but also for mass production of finished products. In 3D printing, parts are usually built in discrete layers. Hence, this manufacturing process results in a certain amount of structural uncertainty in the form of discontinuities, voids, and poor inter-layer bonding. In our previous research, we successfully investigated the differences in the ultimate strength and fatigue life for 3D printed Acrylonitrile Butadiene Styrene (ABS) components built by various build/layer orientations. Our previous research successfully highlighted the ultimate strengths and fatigue life, including SN Curves. However, there is a need for further research to improve the tensile strength and fatigue life of the 3D printed ABS components. This research explores effects of the surface treatment on the tensile strength and fatigue life of the 3D printed ABS components with various layup-orientation. In this study, Acetone Vaper Smoothing (AVS) method was used as the surface treatment of the 3D printed ABS components. Our research found that the AVS method could reduce stress concentrations on the surface and structural uncertainty of the 3D printed ABS components to improve the tensile and fatigue strength. However, these results were occurred after adjusting the Acetone vapor exposure and improving the drying methods because Acetone weakened the layer bonding of the ABS and reduced the tensile strength and fatigue life of the 3D printed ABS components. This research provides the optimal conditions of the Acetone Vapor exposure time and the drying time.
The Fused Deposition Modeling (FDM) is a form of additive manufacturing (AM) in which a material is melted, extruded, and fused in layers to build the intended model, often through the use of 3D printers. Additive manufacturing is a manufacturing process where material is added to form the object rather than removing material or using some form of a mold. Subtractive manufacturing (SM) requires material to be removed to form the desired part utilizing equipment such as cut-ting, milling, grinding, drilling, etc. The requirement to use multiple machines and the expertise to use each one individually is a limitation when using SM for rapid prototyping or other short-run applications. Historically, AM had been primarily used to fill the role of rapid prototyping, but it is expanding and increasingly used to create end user products. The use of AM products has increased design flexibility and speed and is being used to customize products for consumers while reducing waste and the amount of production steps.
Current AM capabilities have limitations due the nature of the fused material being added in layers, which impacts the us-ability for end user products. The products produced by AM methods are not formed from a solid continuous piece of material and so the properties differ based on how the manufacturing process is conducted (i.e., not isotropic). The surface finish is a property of AM that is limited by the layering process. The layers fuse to each other but do not form a continuous and smooth surface finish. Instead, it consists of ridges and valleys of material. Besides the cosmetic drawbacks, such surface
H. Bae (✉) · N. Blair · M. Michaelis · A. Hossain
Department of Mechanical Engineering & Technology, College of Science, Technology, Engineering, and Mathematics, Eastern Washington University, Cheney, WA, USA
S. L. B. Kramer et al. (eds.), Additive and Advanced Manufacturing, Inverse Problem Methodologies and Machine Learning and Data Science, Volume 4, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-50474-7_4
texture is undesirable for product properties such as fatigue life, which is significantly impacted by surface quality and roughness.
Many researchers have investigated the anisotropic behavior of tensile test specimens fabricated by FDM AM techniques [1–6]. However, few researchers have studied the fatigue behavior of ABS specimens fabricated by FDM AM techniques. Ziemian et al. studied the tensile and fatigue behavior of layered acrylonitrile butadiene styrene (ABS) samples, fabricated by FDM method [7]. In that research, typical tensile tests were performed first on FDM specimens with four different layup orientations. Then, the ABS samples were subjected to tension-tension fatigue cycling load to generate SN plots, which were finally used to determine the fatigue strength. SEM images of fracture surfaces of fatigue specimens with different layup orientations were also presented in this research paper. In a study of Bae et al. [8], several fatigue tests were conducted and generated SN curves for the ABS samples with the different layup orientations. The research also calculated and estimated the fatigue strengths of the ABS samples for the different layup orientations.
Dimethyl ketone (acetone) is a solvent of acrylonitrile butadiene styrene (ABS) and is a common secondary process to improve smoothness of ABS printed objects in the hobbyist community. The acetone is vaporized on a heated surface and the model is lowered into the fumes, dissolving the outer surfaces and penetrating into the material based on fume exposure time. After the sample is removed, and given enough time, the acetone evaporates from the material returning it to the original composition but with a visually smoother surface finish.
The application of a chemical post-process to ABS material manufactured with FDM has been the focus of several studies but is still an emerging discussion. Multiple studies have been focused on the effectiveness of acetone vapor being used as a smoothing process for ABS parts made with FDM [9–12]. From these experiments, it was concluded that the treatment was very effective at improving the surface quality. To improve the process, the use of a vacuum chamber was able to reduce the amount of time and solvent needed to achieve the same smoothing effect from the acetone vapor at atmospheric conditions [13]. Concern that the process would impact the geometric accuracy of the parts was the focus of Garg et al. and the authors concluded that cold vapor treatment signi ficantly improved the surface roughness of the part while having minimal impact to the geometric accuracy [14]. Adding a solvent to a material will put it at risk of altering the mechanical properties of the material as well as the surface quality. In a study of vapor smoothing in relation to mechanical strengths, it was determined that the process altered strengths which ruled it out to be used in certain applications, otherwise it was effective at smoothing surface quality [15]. In a study by Cunico et al., the mechanical strengths were tested before and after exposure to a solvent. The peak-peak roughness was greatly reduced when exposed to a correct amount of solvent and it also increased the mechanical strengths [16]. In a study by Neff et al., they concluded that the vapor polishing on thin specimens had minimal impact to mechanical properties but still had vast improvement of surface quality [17]. Vapor polishing is not the only method to improve surface finish, other methods include tumbling, hand finishing, shot peening, etc. ABS immersion in acetone yielded the best geometric accuracy of methods tested [18].
In this study, ABS specimens were manufactured using FDM methods utilizing a 3D printer. The specimens had two unique layup orientations and underwent acetone vapor polishing with two distinct drying methods. After undergoing the vaporization treatment process, the samples were tested in tension and rotational bending fatigue. Conclusions are drawn from the ultimate strength and fatigue life; the latter being estimated by the Stress-Life approach.
Materials and Methods
3D Printing
The fatigue and tensile samples were printed using a Prusa Mk3 3D printer in an enclosure to maintain consistent print conditions. Table 1 shows all the parameters that are used to prepare the 3D printed samples (Fig. 1).
The tensile samples are the standard dog-bone shape samples, as shown in Fig. 2a. The dimensions of these samples are adjusted so that they could fit to our existing tensile testing machine. The fatigue samples are also dimensioned to properly fit the testing machine, but to account for the strength differences between metal and ABS samples, the minimum neck diameter recommended by the fatigue tester manual is increased, as shown in Fig. 2b.
Table 1 3D printing parameters and settings
Tensile and fatigue samples were printed in two variations of the print layup orientation, as shown in Fig. 3. The horizontal print orientation is made by the printer nozzle running parallel to the length axis. If the nozzle is printing normal to the length axis, it is called vertical. The same naming convention applies to the tensile and fatigue samples.
Fig. 1 Prusa i3 MK3 3D printer in its printing enclosure
Fig. 2 Basic dimensioning for (a) tensile samples and (b) fatigue samples (units: inch [mm])
Fig. 3 Tensile samples in the (a) horizontal orientation (b) vertical orientation
Acetone Vapor Treatment Process
To begin the acetone vapor treatment process, the samples were lightly sanded with 320 grit sandpaper to remove any large printing surface defects. In a well-ventilated area, acetone is poured into a glass beaker with a heat safe lid that is non-sealing. The beaker is put on a hot plate set to a medium heat setting. The samples to be treated are held by a copper wire that is bent into a shape that holds the end of the samples by pinching them shown in Fig. 4. When the acetone vapor has risen enough to submerge the entire length of the samples, the samples are lowered into the vapor for 5 s. After the 5 s have passed, the samples are removed and hung up to dry for 5 min.
For the long-term drying process, two different methods were used. The first method was to allow the sample to dry exposed to the open room environment for 3 days. The samples were placed on a paper surface to prevent the material bonding to a hard surface. The other method was in a heated environment produced in a modi fied food dehydrator shown in Fig. 5 The samples were dried at 64 °C for 24 h.
Surface Roughness Measurement
Before and after each sample goes through the acetone treatment process, the surface roughness is measured using a profilometer. The surface roughness was measured at the smallest neck diameter for the fatigue samples and on the thickness direction for tensile samples. The samples roughness was measured along the axial and transverse directions. Figure 6 shows a fatigue sample being measured in the transverse direction.
Tensile Testing
Tensile testing was performed with a Tinius Olsen H50KS tensile testing machine shown in Fig. 7. Testing included both of the orientations and curing conditions, horizontal air-dried, horizontal heated-dry, vertical air-dried, and vertical heated dry. With each configuration, three samples were tested to determine the failure force in tension.
Fig. 4 Acetone vapor treatment process of fatigue samples
Fatigue Testing
Fatigue testing utilized an RFB-200 rotating beam fatigue machine from Fatigue Dynamics. Fatigue test data were gathered for the four different sample configurations: vertical air-dried, vertical heated-dry, horizontal air-dried, and horizontal heateddry. Three samples were tested at each of the five stress amplitudes which translated to 5 in-lb., 4.5 in-lb., 4.0 in-lb., 3.5 in-lb., and 3.0 in-lb. in the machine settings (Fig. 8).
Fig. 5 Food dehydrator used to heat sample to 64 °C. The lid has been removed in the photo to show samples
Fig. 6 Mitutoyo profilometer with a device to hold fatigue samples
Experimental Results
Surface Roughness
The surface roughness was measured before and after the samples were treated with the acetone vapor process. Measurements were taken in two directions because the printing orientation changes which direction the maximum roughness occurs.
Table 2 shows that the chemical vapor smoothing process was able to reduce the overall surface roughness by a minimum of 15 times the highest pre-treatment value. The horizontal prints had an increased surface roughness in the transverse direction because the measurement was going over the layer indents left from the printing process. Inversely, the vertical prints had the largest roughness in the axial direction for the same reasons as the horizontal. The highest pre-treatment roughness values for the vertical and horizontal were all within a 0.753 μm spread. For the post-treatment, the significant reduction of largest pre-treatment roughness values was all within 0.321 μm.
From Figs. 9 and 10, it can be seen that the vapor treatment significantly reduced the visibility of the ridges left from the 3D printing process. The large-scale views of the samples in Fig. 9 show that the vertical print was more uniform before vapor treatment. The more uniform initial surface of the vertical print (Fig. 9b, c) appears to yield a more uniform post treatment surface than the horizontal (Fig. 9e, d). In Fig. 10a, d, the layers are clearly defined and in subsequent images the surface
Fig. 7 Tinius Olsen H50KS tensile tester
Fig. 8 RFB-200 rotational fatigue machine
Table 2 Averaged surface roughness values (Ra) from the four conditions for the fatigue tests (unit: μm)
Fig. 9 Overview images of fatigue samples using a camera to show how the outer surfaces of the samples were altered by the acetone treatment. Top row is of the vertical prints (a) untreated (b) treated and air-dried (c) treated and heated-dry. Bottom row is of horizontal print (d) untreated (e) treated and air-dried (f) treated and heated-dry. (units of scale cm [mm])
appears smooth. The images do not show a clear difference in the surface quality between the two drying conditions. They do appear to show the print direction impacted the surface quality after treatment with the horizontal orientation having a bumpier texture than the vertical print directions, although surface roughness values were comparable. The images support the numerical results gained from the profilometer that the use of acetone vapor is very effective in improving the surface quality of 3D printed ABS objects.
Tensile Testing
The tensile samples were printed in the vertical and horizontal orientations, treated with the acetone vapor treatment, and air-dried or heated-dry. Three samples for each of the four conditions were tested in tensile strength. The tensile test results are shown in Fig. 11.
Fig. 10 Images of Fatigue samples outer surfaces at 50× magnification through the use of a microscope. Top row is the vertical print (a) untreated (b) treated and air-dried (c) treated and heated-dry. Bottom row is horizontal print (d) untreated (e) treated and air-dried (f) treated and heated-dry
Fig. 11 Plots of failure stress for tensile samples: (a) horizontal print (b) vertical print. Each plot has three sets air-dried, heated-dry, and untreated. Untreated samples data was from a previous experiment and the data was aver-aged and plotted as the single average over the three sample trials
From Fig. 11, it can be seen that the failure stress was not substantially impacted by the vapor treatment. The average stress for the horizontal heated-dry samples was 37.5 MPa and was 1.3 MPa higher than the air-dried and 0.3 MPa higher than the untreated samples average. The heated-dry of the vertical print had an average stress of 10.5 MPa that was 0.8 MPa stronger than the air-dried but 0.7 MPa weaker than the untreated samples. The horizontal heated-dry samples were on average 20 MPa stronger than the vertical heated-dry samples. The differences in the samples’ strength are due to the anisotropic properties of 3D printing and it is much stronger when loaded parallel to print orientation. The acetone treatment is a process that starts on the surface and the longer it is in contact with the material the deeper the vapor penetrates. Tensile failure is related to the cross-sectional area. For the process to have an impact on the failure stress, it needs to penetrate most of the thickness. The treatment exposure time was 5 s, which is not long enough for the vapor to seep into the inner material and affect the layer bonding to result in a change in the tensile failure.
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Roma nousi säikähtäen. »Ettehän aikone antaa sitä viranomaisille!» huudahti hän. »Sitä Sir Evelyn tahtoi, mutta minä sanoin hänelle, että jos hän käyttäisi sitä todistuksena erästä toista henkilöä vastaan, kieltäisin kaiken. Paitsi sitä, pyhä isä, se, minkä olen kertonut, on rippisalaisuus, ja ettehän ilmaise sitä kenellekään.»
Paavi ei näyttänyt kuulevan. »Herra, Herra!» huudahti hän hiljaa. »Anna minun syödä tuhkaa leivän asemesta ja sekoittaa juomani kyynelin.»
»Ja sitten, pyhä isä, on vielä jotakin, mitä en sanonut. Hän, joka surmasi ministerin, teki sen puolustaakseen itseään. Se oli vahinko, mutta ellei niin olisi tapahtunut, olisi ministeri surmannut hänet.»
Hän oli polvistunut taas ja kosketti paavin viittaa.
»Minä olen ainoa varsinainen syyllinen, sillä minä olin aikonut sen rikoksen tehdä. Mutta jos hänen kätensä oli välikappaleena ja minä sen tähden saan kärsiä, niin pidän sitä sovituksena. Minähän hänet petin, ja vaikka tein sen rakkaudesta enkä vihasta, ovat seuraukset samat, ja hänen sydämensä on murtunut.»
Paavin valkoinen pää oli hyvin kumarassa.
»Enkä minä voi kärsiä enää kauan. Olen sairas — hyvin sairas — eikä tuskani kestä kauan.»
Hänen suuret, kauniit silmänsä loistivat.
»Ja jos, kuten sanotte, Jumala on kääntänyt kaikki asiat parhain päin, tahtoo Hän ehkä ottaa minut pois jonkun toisen asemesta, jonkun toisen, joka on parempi ja jalompi ja tarpeellisempi.»
Hänen huulensa värisivät, hänen äänensä tukehtui ja hän pyyhki pois kyynelen.
»Näin unta siitä viime yönä, pyhä isä. Luulin, että mieheni oli palannut Roomaan ja kaikki kellot soivat. Se oli unta vain, ettekä te kai usko sellaisiin lapsellisuuksiin. Mutta minusta oli suloista ajatella, että kun en ole voinut elää mieheni hyväksi, voin edes kuolla hänen puolestaan ja siten parantaa rikokseni.»
Paavi ei voinut kestää enempää.
»En saa soimata teitä, tyttäreni. Tulin tänne tapaamaan Maria Magdalenaa, ja minä löysinkin itse Jumalan äidin sielun.»
»Siunatkaa minua, pyhä isä.»
»Minä siunaan teitä, lapseni. Mutta vanha mies, joka on siunannut monta, tarvitsee nyt itse teidän siunaustanne.»
VIII.
Davido Rossi istui koko päivän huoneessaan Vatikaanissa lukien niitä kirjeitä, jotka paavi oli hänelle antanut.
Ne olivat samat kirjeet, jotka Roma oli lähettänyt Lontooseen, Pariisiin ja Berliiniin.
Hän luki ne yhä uudelleen, ja paitsi kellon naksutusta ei tuossa suuressa huoneessa kuulunut muita ääniä kuin Rossin syvät huokaukset. Mitä erilaisimmat tunteet risteilivät hänen sydämessään, ja hän tuskin ymmärsi, oliko hän iloinen vai surullinen siitä, että näin myöhään, kun kova kohtalo jo oli tehnyt tehtävänsä, nuo rauhan sanomat olivat saapuneet hänelle.
Noista hienoista, läpikuultavista paperilevyistä näytti huokuvan esiin henki, joka tunkeutui koko hänen olemuksensa läpi. Kun hän luki nuo sanat, joskus iloiset, joskus surulliset, milloin täynnä riemua, milloin tuskasta valittavat, kohosi eloon kokonainen maailma helliä tunteita, jotka poistivat kaikki mustat intohimot.
Hän saattoi nähdä itse Roman, ja hänen sydämensä sykki kuin ennen Roman suloisen, sanomattoman tenhovoiman alla. Nuo rakkaat kasvot, nuo ihanat silmät, tuo ääni, tuo hymy — ne
ilmestyivät taas hänelle kiduttaen häntä rakkauden ja katumuksen kidutuksin.
Kuinka uljaasti hän oli vastustanut vihollisia! Tuo nuori, tulinen, viehättävä, onnellinen olento oli uhrattu tuskalle. Ja tuo hänen liikuttava salaisuutensa — kuinka suloisesti ja rehellisesti hän oli sen kertonut! Ainoastaan puhdas ja uljas nainen saattoi menetellä niin. Mutta hän, Davido Rossi, oli silmittömässä vihassaan ja mielettömässä raivossaan käyttäytynyt kuin alhainen tyranni ja pelkuri. Kuinka suurenmoisesti hän oli vakuuttanut rakkauttaan, kun hän luuli tuon loukkauksen koskevan toista miestä! Mutta kun hän sai tietää, että se koski häntä itseään, kuinka nopeasti hän oli muuttunut, ja hän oli kironnut vaimoraukkaa, joka oli luottanut häneen!
Mutta rakkautta ja katumusta voimakkaampana Rossia vaivasi ääretön epätoivo. Se ilmeni kapinana Jumalaa vastaan, joka oli sallinut sokean, julman kohtalon särkeä kahden syyttömän lapsen onnen. Kun hän haki turvaa Vatikaanista, oli hänellä kai pieni toivon kipinä jälellä. Se oli sammunut nyt, eikä maksanut vaivaa pyrkiä enää. Ihmisen mitättömyys taistelussa kohtalon kovuutta vastaan teki kaiken pyrkimisen mahdottomaksi.
Rossi oli soittanut kelloa pyytääkseen päästä hänen pyhyytensä puheille, kun ovi aukeni ja paavi itse astui sisään.
»Pyhä isä, olisin tahtonut puhutella teitä.»
»Mitä tahdotte, poikani?»
»Olisin tahtonut puhua itsestäni. Nyt ymmärrän, kuinka väärin tein pyytäessäni suojaa teiltä. Te luulitte minua viattomaksi, mutta minä
en kertonut teille kaikkea. Kun sanoin olevani syyllinen Jumalan ja ihmisten edessä, ette ymmärtänyt mitä tarkoitin. Pyhä isä, tarkoitin että olin murhaan vikapää.»
Paavi ei vastannut, ja Rossi jatkoi, ääni väristen tuskallisista tunteista, jotka täyttivät hänen sydämensä.
»Totta puhuen, teidän pyhyytenne, en ole ajatellut sitä itsekään. Tekoni tein osittain henkeni puolustukseksi enkä pitänyt sitä rikoksena. Ja se, jonka surmasin, oli kurja mies, jonka sydän oli täynnä ilkeyttä, enkä minä tuntenut enempää katumusta hänen kuoltuaan, kuin jos olisin astunut myrkyllisen käärmeen päälle. Mutta nyt ajattelen asiaa toisella lailla. Tuloni saattoi teidät vaaraan. Pyydän anteeksi teiltä ja tahdon mennä pois nyt.»
»Minne tahdotte mennä?»
»Minne tahansa — en tiedä vielä.»
Paavi katsoi uurtuneita nuoria kasvoja, jotka kuvastivat pelkkää epätoivoa, ja hänen sydämensä suli.
»Istukaa, poikani. Ajatelkaamme. Vaikk'ette kertonut minulle murhasta, sain kohta tiedon siitä… Puolustaaksenne henkeänne, niinkö?»
»Niin, mutta en tahdo sillä puolustautua. Ja jos koettaisinkin puolustautua, ei kukaan muu sitä voi todistaa.»
»Eikö kukaan?»
»Ehkä yksi. Mutta se on vaimoni, eikä hän välittäne pelastumisestani enää, vaikka tahtoisinkin pelastua… Olen lukenut
hänen kirjeensä.»
»Jos kertoisin teille, että asia ei ole niin — että vaimonne vieläkin on valmis uhrautumaan teidän tähtenne…»
»Mutta se on mahdotonta, teidän pyhyytenne. On paljon semmoista, mitä ette tiedä.»
»Jos kertoisin teille, että olen juuri tavannut hänet ja huolimatta siitä, että te ette luota häneen, hän yhä vielä luottaa teihin…»
Epätoivon ilme alkoi poistua Rossin kasvoista ja hän huudahti ilosta.
»Jos kertoisin, että hän rakastaa teitä ja on valmis antamaan henkensä teidän puolestanne…»
»Onko se mahdollista? Sanotteko niin? Huolimatta kaikesta! Ja missä — missä hän on? Antakaa minun mennä hänen luokseen.
Pyhä isä, jos tietäisitte —! Minä tahdon mennä pyytämään häneltä anteeksi. Minä kirosin häntä! Se on totta, että sokeassa, hurjassa raivossani minä… Mutta antakaa minun mennä hänen luokseen polvillani rukoilemaan anteeksi. Koko lopun elämääni tahdon käyttää parantaakseni, mitä olen häntä vastaan rikkonut.»
»Viipykää, poikani. Saatte nähdä hänet kohta.»
»Onko mahdollista, että saan nähdä hänet? Luulin, etten koskaan enää saisi häntä tavata, mutta Jumala määräsi toisin. Ah! Jumala on hyvä kuitenkin! Ja pyhä isä on hyvä myöskin. Hän on antava minulle anteeksi. Sitten me pakenemme jonnekin — Afrikkaan, Intiaan, minne tahansa. Me ryöstämme pari onnen vuotta, mitä muuta ihminen voi toivoa tässä maailmassa?»
Kiihoittuneena tuota ajatellessaan Davido Rossi näytti unohtavan kaiken — rikoksensa, työnsä, kansansa.
»Onko hän vielä kotona?»
»Hän on vain muutaman askeleen päässä täältä, poikani.»
»Muutaman askeleen päässä! Oi, antakaa minun mennä heti. Missä hän on?»
»Angelon linnassa», vastasi paavi.
Synkkä pilvi pimitti Rossin kasvot, ja hän huudahti hämmästyksestä.
»Siis… vankilassa.»
Paavi nyökäytti päätään.
»Miksi?»
»Pääministerin murhasta.»
»Roma?… Mikä hullu minä olin, kun en ajatellut, että niin saattoi tapahtua! Jätin hänet sen kuolleen miehen luo. Ja kuka uskoo häntä, vaikka hän kieltää surmanneensa paronin?»
»Hän ei sitä kieltänytkään. Hän myönsi.»
»Myönsi? Hän sanoi siis surmanneensa…»
Paavi nyökkäsi taas.
»Siis… siis… hän tahtoi siten pelastaa minut?»
»Niin.»
Rossin silmät kyynelehtivät. Hän oli kuin toinen mies.
»Mutta… tuomioistuin ei voi häntä uskoa.»
»Hänet on jo tuomittu.»
»Tuomittu? Sanoitteko niin? Rikoksesta, jota hän ei ole tehnyt! Ja pelastaakseen minut! Pyhä isä, uskotteko, että viimeinen sanani hänelle… Mutta hän on enkeli. Viranomaiset ovat hulluja. Eikö kukaan epäillyt minua. Eikö kukaan tiennyt, että olin ollut siellä sinä yönä?»
»Yksi ainoa todistuskappale liitti teidät tuohon rikokseen, poikani. Se oli tämä.»
Paavi veti esiin vangitsemiskäskyn, jonka hän oli ottanut Romalta.
»Hänelläkö se oli?»
»Niin.»
Rossin liikutus kasvoi myrskyksi. Epätoivo, joka äsken oli hänet vallannut, näytti pieneltä ja mitättömältä mahtavan tunteen rinnalla. Mutta vähän ajan perästä hänen kosteat silmänsä alkoivat loistaa.
»Pyhä isä, tämä paperi on minun, ja teidän täytyy antaa se minulle.»
»Mitä aiotte tehdä, poikani?»
»Ei ole muuta kuin yksi tehtävä.»
»Mikä se on?»
»Pelastaa hänet.»
Ei tarvinnut kysyä kuinka. Paavi ymmärsi, ja hänen rintansa sykki ja paisui. Mutta nyt, täytettyään tehtävänsä, herätettyään nukkuvan sielun ja annettuaan sille toivoa ja uskoa ja rohkeutta astua oikeuden eteen, vaikkapa kuolemankin eteen, paavi tunsi äkkiä omassa sydämessään jotakin, mitä hän turhaan koetti tukahduttaa.
»Olkoon kaukana minusta rikoksen puolustaminen, poikani, mutta armollinen Jumala, joka käyttää meidän tunteitammekin omiin suuriin tarkoituksiinsa, on antanut tekonne loppua hyvin. Maailma vapisee tuntemattomien tapahtumien kynnyksellä, eikä kukaan tiedä mitä huomispäivä voi tuoda mukanaan. Odottakaamme hiukan.»
Rossi pudisti päätään.
»Totta on, että rikos on sama huomenna kuin tänään, mutta se kuollut mies oli tyranni, verenhimoinen tyranni, ja jos hän pakotti teidät oman henkenne puolustukseksi…»
Taas Rossi pudisti päätään, mutta paavi jatkoi.
»Teidän täytyy ajatella omaa elämäänne, poikani, ja kuka tietää, vaikka Jumala tahtoisi…»
»Antakaa minun mennä.»
»Aiotteko ilmaista itsenne poliisille?»
»Aion.»
Paavi ei virkkanut enää mitään. Hän nousi seisomaan. Hänen pyhimyskasvonsa osoittivat sanomatonta rakkautta ja ylpeyttä. Hän ajatteli vuosia, jolloin hän turhaan oli etsinyt poikaansa ja jotka nyt olivat päättyneet tähän yhtymiseen ja eroamiseen, ja hän olisi tahtonut pusertaa tuota nuorta miestä sykkivää sydäntänsä vastaan. Mutta hetken perästä sanoi hän entisaikoja muistuttavalla tyynellä, uljaalla äänellä:
»En hämmästy päätöstänne, poikani Se on nimenne ja sukunne arvoinen. Ja nyt, kun eroamme viimeisen kerran, tahtoisin kertoa teille jotakin.»
Davido Rossi ei vastannut.
»Minä tunsin äitinne, poikani.»
»Äitini?»
Paavi nyökäytti päätään ja hymyili.
»Hän oli uljas sielu ja sai kärsiä äärettömästi. Semmoiset ovat Jumalan tiet.»
Davido Rossi ei puhunut. Hän katsoi paavin väriseviä kasvoja ja koetti pysyä tyynenä.
»Tietysti ajattelette pahaa isästänne tietäessänne, kuinka paljon äitinne sai kärsiä. Eikö niin?»
Rossi kohotti toisen kätensä otsalleen ikäänkuin tyynnyttääkseen ajatuksiaan ja sanoi: »Kuinka minä saisin ajatella pahaa kenestäkään, minä, joka olen saattanut oman vaimoni kärsimään?»
Paavi hymyili taas arasti.
»Davido…»
Rossi ei voinut hengittää.
»Jos Jumala sallisi sinun tavata isäsi jossakin ja hän ojentaisi sinulle kätensä, tahtoisitko… olkoonpa hän missä tahansa ja kuka tahansa… tahtoisitko puristaa hänen kättään?»
»Tahtoisin», sanoi Rossi, »vaikkapa tapaisin hänet vankilassa ja vaikkapa hän olisi kurjista kurjin.»
Paavi hengitti syvään, astui askeleen eteenpäin ja ojensi ääneti kätensä. Seuraavassa silmänräpäyksessä Davido Rossi ja vanha paavi seisoivat käsi kädessä ja silmä silmää vastaan.
He koettivat puhua, mutta eivät voineet.
»Hyvästi!» sanoi paavi puoleksi tukehtuneella äänellä ja astui vavisten pois huoneesta.
IX.
Koko päivän Vatikaanissa oli ollut sama tunnelma kuin laivassa, joka pyrkii pakoon myrskyn alta. Papit kuiskailivat kalpeina toisilleen, sveitsiläisen kaartin upseerit vaihtoivat kiireellisiä sanoja, ja sanansaattajat kulkivat yhtämittaa majordomon ja maestro di cameran virastojen välillä. Myrsky uhkasi taivaanrannalla, merimiehet olivat kannella, mutta kapteeni viipyi alhaalla.
Vasta Angeluksen jälkeen paavi suostui vastaanottamaan kardinaalisihteerin. Vihdoin kardinaali oli lähettänyt melkein käskevän sanan. Asia oli hyvin tärkeä. Oli mahdotonta enää viivytellä.
Kun kardinaalisihteeri astui paavin huoneeseen, näytti hänen suuri, kömpelö vartalonsa mustassa, punaisella kirjaillussa kaavussa vähemmän taipuisalta kuin ennen, hänen äänensä oli vähemmän nöyrä, hänen käytöksensä vähemmän kohtelias kuin ennen.
»Mitä nyt, teidän kunnianarvoisuutenne?» kuului paavin väsynyt ääni. Hän istui pienen uunin edessä, valkoinen viitta hiukan kohotettuna, ja lämmitti tohveleihin pistettyjä jalkojaan sinertävän tulen edessä.
Kardinaalisihteeri selitti. Kello kymmenestä saakka hänen oli täytynyt kysymättä hänen pyhyytensä neuvoa hoitaa erittäin tärkeätä kirjeenvaihtoa valtion viranomaisten kanssa. Tuntemattomalla tavalla viranomaiset olivat saaneet tietää, että eräs henkilö, joka oli vangitsemiskäskyn alainen, oli saanut suojaa Vatikaanissa, ja koska oli vaarallista yleiselle turvallisuudelle, että tämä mies oli vapaana, vaativat viranomaiset paavia heti paikalla antamaan hänet heidän käsiinsä.
»Ja mitä te vastasitte?»
»Minun vastaukseni oli se, että me emme tunnusta, että kysymyksessä oleva henkilö on meidän alueellamme, mutta jos hän olisi siellä, antaa Italian hallituksen lupaus pyhän istuimen suojelemisesta Vatikaanille oikeuden olla erityisenä valtiona, jolta ei mitenkään voi vaatia valtiollisen pakolaisen luovuttamista.»
»Ja mitä he vastasivat?»
»He vastasivat, teidän pyhyytenne, että turvalain 17:s pykälä selvästi kieltää Vatikaanilta oikeuden sellaisiin toimenpiteisiin, jotka sotivat yleistä järjestystä vastaan, ja paenneen rikoksellisen suojeleminen saattaisi paavin rikoslain alaiseksi.»
»Mitä te siihen sanoitte?»
»Minä sanoin, että turvalaki, jos sillä on mitään merkitystä, antaa paaville hallitsijan oikeudet, eikä hän hallitsijana voi olla toisen maan lakien alainen.»
»Entä sitten?»
»Sitten, teidän pyhyytenne, viranomaiset lähettivät varoituksen, että jos kysymyksessä olevaa henkilöä, jonka tiedetään olevan Vatikaanissa suojassa, ei toimiteta ulkopuolelle Vatikaanin rajoja päivän kuluessa, täytyy valtion, vaikka vastenmielisesti, lähettää väkeä pakolla vangitsemaan hänet.»
»Ja teidän vastauksenne?»
»Minä vastasin, että 7:s pykälä selvästi kieltää sotilaita astumasta Vatikaaniin ilman paavin lupaa ja että paavi kieltää sen ja aikoo vastustaa sitä.»
»Ja mitä he vastasivat?»
»Heidän vastauksensa… suvaitseeko teidän pyhyytenne katsoa ikkunasta?… Heidän vastauksensa siihen on rykmentti jalkaväkeä Borgon kasarmeista ja komppania tykistöä sekä yksi kanuuna.»
Kardinaalisihteeri ei voinut säilyttää tavanmukaista kohteliaisuuttaan, vaan puhui karkealla äänellä ja kulki levottomasti edestakaisin.
»Teidän pyhyytenne suvaitkoon muistaa, että varoitin teitä. Jos, silloin kun tuo mies lausui julkean toivomuksensa saada suojaa, teidän pyhyytenne… Mutta teidän pyhyytenne puhe oli kaiken pahan alkuna. Se todisti jonkinmoista heikkoutta, jota hallitus kiirehti käyttämään hyväkseen, ja nyt…»
Paavi teki kärsimättömän liikkeen.
»Meidän täytyy toimia nopeasti, teidän pyhyytenne. Sotamiesten saapuessa sveitsiläinen kaarti sulki pronssiportin, minkä jälkeen
piirittäjien kapteeni lähetti sanan, että ellei porttia avata tunnin kuluessa, hän antaa tykkiväen ampua sitä.»
Paavi nousi uunin äärestä. Vihdoinkin hän oli herännyt. Katsoen ikkunasta hän näki sotamiehet piazzalla. Sitten hän mutisi itsekseen astuessaan lattialla: »Oi Sinä, jonka kädessä maailman kohtalot on… Sinun tahdostasi me hajaannumme kuin kärpäset myrskyssä…
Mitä minun pitää tehdä? Mitä minun nyt tulee tehdä?»
»Jos teidän pyhyytenne kysyy minulta mitä teidän pitää tehdä, sanon — ei mitään.»
»Ei mitään?»
»Antaa heidän ampua rikki pronssiportti. Se on ainakin osoittava katolilaisille ja katolisille valtioille, kuinka mahdoton paavin ja hänen ministeriensä nykyinen asema on.»
»Te neuvotte tekemään vastarintaa?»
»Niin paljon vastarintaa kuin tarvitaan, jotta maailma näkisi, että me olemme väkivallan uhreja.»
»Ja veri, joka sitten ehkä vuotaa…»
»Se… teidän pyhyytenne… jos niin onnettomasti kävisi…»
Muutamia minuutteja kului. Paavin oven edessä oli kiihkeätä, vaikka äänetöntä liikettä. Sotilaat asettuivat paikoilleen ladatut revolverit ja paljastetut miekat kädessä. Huoneessa oli kiihkeä tunnelma. Maestro de camera tuli sanomaan, että puutarhapaviljonki oli järjestetty paavia varten. Sitten majordomo ilmoitti, että paavin vaunut olivat Fondamentan ovella ja että Santa Monican puistokäytävä oli pimeä ja tyyni. Vihdoin Cortis levitti mustan viitan, jota paavi oli käyttänyt käydessään Angelon linnassa, ja puhui jotakin paosta.
Paavi oli palannut uunin ääreen peittäen valkoista patalakkia käsillään, kun piazzalta kuului kova torvien toitotus. Se oli järjestyksen valvojan merkki. Paavi tiesi, mitä se tarkoitti, ja nousi seisomaan. Vakavin askelin hän astui ovelle ja avasi sen. Soturit ulkopuolella hämmästyivät sanattomiksi.
»Hyvät herrat», sanoi paavi, »jos tahdotte seurata meitä, käskemme teitä panemaan pois aseenne.»
»Teidän pyhyytenne!» huudahtivat kaartin upseerit, mutta tottelivat heti.
Paavi kulki pitkien käytävien läpi sotilaitten edellä. Hän astui alas portaita suureen saliin. Komppania sveitsiläistä kaartia seisoi aseilla varustettuna pronssiportin edessä, joka oli suljettu rautatangoilla ja pölkyillä.
»Hyvät herrat», sanoi paavi, »me käskemme teitä heittämään pois pistimenne ja pyssynne.»
»Teidän pyhyytenne!» huudahtivat soturit, mutta hekin tottelivat.
»Avatkaa nyt portti», sanoi paavi.
»Teidän pyhyytenne!»
»Avatkaa se!»
Pronssiportti avattiin.
Ulkopuolella seisovalla kapteenilla oli täysi työ rohkaista miehiään. Taistellen italialaista taikauskoaan vastaan he näyttivät levottomilta ja epäluotettavilta. Kerran he luulivat näkevänsä paavin ikkunan läpi ja olivat huutamaisillaan eläköötä hänelle. Aselepo ei tuntunut koskaan loppuvan, päivä laski ja pimeys kävi yhä synkemmäksi.
Mutta vihdoin järjestyksen valvoja antoi merkin ja kapteeni lausui komentosanan. Silloin miehet liittyivät riviin ja marssivat pronssiportille.
Sillä hetkellä se avattiin.
»Valmiit!» huusi kapteeni, ja miehet tarttuivat aseisiin valmistautuakseen rynnäkköön sisältäpäin.
Seuraavassa silmänräpäyksessä valkoinen olento ilmestyi avonaiselle portille lyhtyjen valaistessa hänen kalpeita, kuluneita kasvojaan. Se oli paavi. Pyssyt putosivat miesten olkapäiltä ja he huokasivat syvään.
»Sotilaat», sanoi paavi, »miksi olette tulleet tänne? Ampumaan Vatikaania? Ovi on auki ja te saatte astua sisään. Miksi teillä on pyssyt ja kanuunat? Teitä ei vastusta kukaan muu kuin heikko, vanha mies.»
